A transmitting antenna at the top of a tower has a height of $h_T=32\text{ m}$ and the height of the receiving antenna is $h_R=50\text{ m}$. Given the radius of the Earth $R=6400\text{ km}$, calculate the maximum line-of-sight distance $d_m$.

Solution

1. The formula for the maximum line-of-sight distance is $d_m=\sqrt{2R{h}_T}+\sqrt{2R{h}_R}$, where $$\begin{array}{rl}R& =6400\text{ km}\\ & =6.4\times 10^6\text{ m}\end{array}$$, $h_T=32\text{ m}$, and $h_R=50\text{ m}$.
2. Calculate the individual distances: $$\begin{array}{rl}{d}_T& =\sqrt{2\times 6.4\times 10^6\times 32}\\ & =\sqrt{409.6\times 10^6}\approx 20200\text{ m}\\ & =20.2\text{ km}\end{array}$$.
3. Calculate the receiving distance: $$\begin{array}{rl}{d}_R& =\sqrt{2\times 6.4\times 10^6\times 50}\\ & =\sqrt{640\times 10^6}\approx 25300\text{ m}\\ & =25.3\text{ km}\end{array}$$.
4. Sum the distances to find the total: $$\begin{array}{rl}{d}_m& =20.2\text{ km}+25.3\text{ km}\\ & =45.5\text{ km}\end{array}$$.
5. Hence the answer is (A).